Quasi-Trefftz spaces for a second order time-harmonic Maxwell's equation

Abstract

Quasi-Trefftz methods are a family of Discontinuous Galerkin methods relying on equation-dependent function spaces. This work is the first study of the notion of local Taylor-based polynomial quasi-Trefftz spaces for a system of Partial Differential Equations (PDEs). These discrete spaces are meant to discretize a quasi-Trefftz weak formulation, and are introduced here for electro-magnetic wave propagation in inhomogeneous media, governed by a second order formulation of Maxwell's equation with variable coefficients. Thanks to an adequate Helmholtz decomposition for spaces of homogeneous polynomial vector fields, the outcome is the proposed quasi-Trefftz spaces with same best approximation properties as the corresponding standard polynomial spaces but with an optimal dimension, as well as a procedure to construct a quasi-Trefftz basis.